For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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48 views

Linear Algebra Matrix algebra

Let $S: \mathbb R^3 \rightarrow \mathbb R$ $v= (v_1,v_2,v_3) w= (w_1,w_2,w_3)$ Both w and v are vectors Express the standard matrix $S: \mathbb R^3 \rightarrow \mathbb R$ in terms of ...
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0answers
32 views

Nil and nilpotent restricted lie algebras

Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here. ...
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1answer
110 views

Solution to this linear equation system

So this is my homework : Let $$ A= \begin{bmatrix} 1 & 0 & 1 & 3 \\ 2 & 0 & \lambda & 6 \\ 1 & 1 & 1 & 1 \\ \end{bmatrix} ...
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24 views

Homogeneous matrix to represent rotation out of plane?

How to represent 2D homoheneous projection matrix $M=\left( \begin{array}{ccc} m_{00} & m_{01} & m_{02} \\ m_{10} & m_{11} & m_{12} \\ m_{20} & m_{21} & w \end{array} \right)$ ...
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1answer
508 views

Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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1answer
93 views

Getting rotation matrix from a vector

I have a vector pointing in some direction and I'm trying to find a matrix $M$ that rotates the vector $v_1=(1,0,0)$ to $v_2=(x,y,z)$, i.e., $M v_1 = v_2$. What is $M$ if $v_1$ and $v_2$ are known? ...
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1answer
442 views

Finding a matrix where the column space is a subset of the null space [duplicate]

Let $A_{3x3}$ be a matrix $ \ne 0$ such that the column space of $A$ is a subset of the null space of $A$. I need to find $A$. Here's my process so far: let $v_1, v_2, v_3$ be the column vectors of ...
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3answers
78 views

How to prove that $\|e^{X+Y}-e^X\| \leq \|Y\| e^{\|X\|} e^{\|Y\|}$?

A couple of questions from the Wikipedia "matrix exponential" article: In the part of the article I linked to, they mention that to conclude that every matrix in $GL(n)$ has a logarithm (though not ...
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1answer
44 views

Which type of matrix has this property D F = F^{-T}

I was given a hint that D det F = det F F^{-T} where D is Frechet derivative that is total derivative. by chain rule. Why does the following hold? For which ...
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43 views

Trace and eigenvalues

In the proof of Proposition 24 of this paper, the following statement is made: "The inequality $Tr(\xi)^2 \geq 1 − \gamma$ implies that the maximum eigenvalue of the Hermitian matrix $\xi$ is at ...
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1answer
54 views

Egienvalues of a product of symmetric matrices

Let $\boldsymbol\Gamma$ and $\boldsymbol W$ be two real $K\times K$ matrices defined as $$ \boldsymbol\Gamma= \left(\begin{array}{cccc} \frac{1}{\gamma_1}& 0 & ...&0\\ & & \\ ...
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3answers
165 views

Chain rule for matrix exponentials

I need help in proving the following theorem: If $M(t)$ is an $n \times n$ matrix of differentiable functions, then $$ \frac{d}{dt}\left( \exp(M(t))\right) = \frac{d}{dt}M(t) \exp(M(t)) = ...
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2answers
114 views

On $n\times n$ matrices $A$ with trace of the powers equal to $0$

Let $R$ be a commutative ring with identity and let $A \in M_n(R)$ be such that $$\mbox{tr}A = \mbox{tr}A^2 = \cdots = \mbox{tr}A^n = 0 .$$ I want to show that $n!A^n= 0$. Any suggestion or ...
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1answer
60 views

If a and b are matrices then what does a|b stand for?

Given $a =(1, 2, 3)^T$ and $b = (5, 8, 4)^T$ what is $C=(a|b)$ mean? is this a given $b$? or a over $b$? how to work out $C$?
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69 views

Are solutions of $\frac{1}{2}(A^T+A)x=b$ and $Ax=b$ related?

I saw some statements about these 2 systems while I was reading something about linear algebra. So I am curious if the solutions of these 2 systems are related. If it is, how are they related? Thanks ...
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1answer
1k views

QR-factorisation using Givens-rotation. Find upper triangular matrix using Givens-rotation.

I'm looking into QR-factorisation using Givens-rotations and I want to transform matrices into their upper triangular matrices. I know how to do this for matrix $ B \in \mathbb{R}^{m\times m}$ but how ...
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1answer
51 views

Combine 2 sparse QR factorizations

I have sparse matrix $A_1$ which is size $m_1 \times n$ and another sparse matrix $A_2$ which is size $m_2 \times n$, where $m_1 < n$ and $m_2 \leq n$ and plan on stacking them to make a sparse ...
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1answer
109 views

Why Gaussian method is recommended for $4\times4$ determinant?

I wanted to know why Gaussian elimination method in Linear Algebra has order of $n^3$ for calculation and found this.But I don't know why this: From the table of $n^3$ vs $n!$ we see that $4^3 ...
5
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2answers
76 views

Do I influence myself more than my neighbors?

Consider relations between people is defined by a weighted symmetric undirected graph $W$, and $w_{ij}$ shows amount of weight $i$ has for $j$. Assume all weights are non-negative and less than $1$ ...
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1answer
206 views

Direction Cosines and Rotation Angles

I'm rotating an object in 3D space with respect to a relative base, or reference frame. I'm using a normal vector to represent the rotation angles. Suppose you have an object parallel to the ...
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1answer
67 views

Orthogonal projection on span $x$

If $x = (2,1)$ and $y = (1,-1)$ how can I find orthogonal projection on the span of $x$? And projection on the span of $x$ along the span of $y$? What I have done is, for the first question, I did ...
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2answers
330 views

Unitary matrix with all its eigenvalues equal to 1 must be the identity matrix

Let $A$ be a unitary matrix over field $F$ ($F=\mathbb{R},\mathbb{C}$) Prove that if all its eigenvalues equal to $1$ then $A$ must be the identity matrix $I$. I am having a hard time figuring out ...
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1answer
163 views

Matrix over a finite field?

I am trying to solve the following problem: Given is a $3\times 3$ matrix $M$ over $\mathbb{F}_{7}$, such that for every vectors $v,w\in \mathbb{F}_{7}^3\setminus \{0\}$ there exists an integer $n$ ...
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2answers
74 views

induction proof of a determinant $n \times n$

I have to proof the following property: Can somebody help my with a few steps for n=n+1? Thanks in advance. Cheers.
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1answer
61 views

Formulating regression model in matrix notation

The observations $y_1, y_2, y_3$ were taken on the random variables $Y_1, Y_2, Y_3$ where $Y_1=\theta+e_1$ $Y_2=2\theta - \phi+e_2$ $Y_3=\theta +2 \phi+e_3$ and $E(e_i)=0, var(e_i)=\sigma^2 ...
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1answer
33 views

homework - Show a matrix as a combination of other matrices and long division

The topic we are dealing with here is polynomial division. The question is: We are given a polynomial: $f(x) = (x+1)(x-1)^2$, and a matrix $D \in R^{nxn}$ such that $f(D)=0$ Using only $I, D, D^2$ ...
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2answers
63 views

Given $A, B\in R^{n\times n}$ diagonal matrices, there exist $p,q \in R[x]$ and $X\in R^{n\times n}$ such that $A = p(X),B=q(X)$

(1) We are given $A,B \in R^{n\times n}$ diagonal matrices of n rows and n columns with real values. Show that there are $X \in R^{n\times n}$ and polynomials $q$ and $p$ such that: ...
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1k views

Find the normalized eigenvectors of a matrix

Let $T=\begin{pmatrix}5 & 0 & 0 \\ 0 & 2 & i\\ 0 & -i & 2 \end{pmatrix}$. I found the eigenvalues and eigenvectors already and they are $1,3,5$ and ...
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1answer
140 views

Are there sometimes only finitely many square roots of a positive matrix?

Question: A positive (semi-) definite matrix has a unique positive (semi-) definite square root. What are its other square roots? In some cases there are infinitely many (such as for $aI$). Are there ...
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3answers
59 views

For which t is the matrix invertible?

$$\begin{matrix} t&a_2&0&0&\cdots&0\\ 0&t&a_3&0&\cdots&0&\\ \vdots&\vdots&\ddots&&\cdots&\vdots\\ ...
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0answers
35 views

Decomposition of a symmetric semi-definite matrix into sums of sparse symmetric semi-definite matrix

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
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4answers
121 views

Show $\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A,A\in M(2,\mathbb{C})$

Show $$\exp(A)=\cos(\sqrt{\det(A)})I+\frac{\sin(\sqrt{\det(A)})}{\sqrt{\det(A)}}A$$ for $A\in M(2,\mathbb{C})$. In addition, $\operatorname{trace}(A)=0$. Can anyone give me a hint how this can ...
3
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4answers
791 views

If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A.

Prove the following statement: If $A^2+A=0$,then $\lambda=1$ cannot be an eigenvalue of A. I've been struggling on this question for a couple of hours and don't know how to approach it.
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2answers
137 views

Prove or disprove: if $A$ is nonzero $2 \times 2$ matrix such that $A^2+A=0$, then A is invertible

if $A$ is nonzero $2 \times 2$ matrix such that $A^2+A=0$, then A is invertible I really can't figure it out. I know it's true but don't know how to prove it
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1answer
55 views

Find a similar matrix with some conditions

Can you find a matrix $B$ which is similar to $$A=\begin{pmatrix} 0 & {-1} & {-1}\\ 1 & 2 &1\\ {-1} & {-1} & 0 \end{pmatrix}$$ such that $(1,0,1)\in \ker(B)$ and ...
2
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0answers
43 views

Parametric QR factorization: $\mathbf{D}(\alpha)\mathbf{V}^*$ a diagonal times a constant unitary matrix

Given a constant unitary matrix $\mathbf{V}^*$ and a parameter diagonal matrix $\mathbf{D}$, can the QR factorization of many different $\mathbf{D}\mathbf{V}^*$ be performed efficiently? Here ...
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1answer
96 views

Plot of ||X||infinity norm

Can anybody tell me why the plot of $\|X\|_{\infty}$ in $\mathbb{R}^2$ comes out to be square? Since $\|(x_1,x_2)\|_{\infty} = \max\{|x_1|,|x_2|\}$, then let us say $|x_1|$ is max. Why the plot is ...
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1answer
62 views

Solving a Gaussian elimination problem.

I have been given a graph with n nodes. Now, I have to color every node of this graph by k colors, number from 0 to k-1. Now, there is a rule. For a node $x$ with adjacent nodes $y_1 , y_2, y_3, ...
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1answer
52 views

Isolate for matrix X

So I have matrices A, B, X, C, D. Isolate for matrix X. $(A+B)XA^2+C=D$ I'm pretty stuck due to the squared part of this problem. Help appreciated!
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1answer
52 views

Generic element of Kronecker product

How can we find the generic element of the Kronecker product of two matrix: Let $A=(a_{ij})_{{1\leq i\leq n}, {1\leq j\leq p}}$ and $n\times p$ matrix and $B=(b_{ij})_{{1\leq i\leq m}, {1\leq j\leq ...
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0answers
46 views

Finding Steady state using markov chains. Am I right?

Suppose that there are two doctors in a country town, Dr Black and Dr White. Each year, 13% of patients move from Dr Black to Dr White, while 19% of patients move from Dr White to Dr Black. Suppose ...
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2answers
564 views

How to show, that a Hermitian matrix is positive definite, if all eigenvalues are positive.

I want to show, that a Hermitian matrix is positive definite, if all eigenvalues of the matrix are positive. And the other way round. Also I wonder, if every Hermitian, strict diagonally dominant ...
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1answer
222 views

Nilpotent matrix and Jordan form

Could you help me solve this problem? Give an example of two nilpotent matrices $N_1$, $N_2$ $ \in M_{n,n} (\mathrm{F})$ with $N_1N_2 = N_2N_1$ such that there is no matrix B with $B^{-1}N_1B$ and ...
2
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0answers
59 views

Ultrametric matrices and their inverse

A non-negative square matrix $A$ is ultrametric iff: $A(i,i)>\{A(i,k),A(k,i)\}\forall k,i$ $A(i,j)\geq\min\{A(i,k),A(k,j)\}\forall i,j,k$ It is well-known that the inverse of non-negative ...
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46 views

A formula for functions of diagonalizable matrices

In one journal article that I have read recently, they use the following formula for computing functions of diagonalizable matrices: $$f(A)_{ij} = \sum_{\lambda \in \textrm{sp}A} \frac{x_i(\lambda) ...
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2answers
67 views

Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$

Given two matrices $A$ and $B$. How would one prove that the Kronecker product of $A$ and $B$ is similar to the Kronecker product of $B$ and $A$?
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1answer
117 views

Order of some matrices in $GL(2,p)$ is coprime with $p$

Let $M$ belongs to $GL(2,p)$ where $p$ is a prime number, and $\det M$ generate $GL(1,p)$, so I want to prove that the order of $M$ is coprime to $p$. I think if $M^{np}=I_2$ that means ...
2
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3answers
455 views

Find the inverse of a matrix with a variable

$$X= \begin{pmatrix} 2-n & 1 & 1 & 1 & \ldots & 1 & 1 \\ 1 & 2-n & 1 & 1 & \ldots & 1 & 1 \\ \vdots & \vdots & \vdots & \vdots & \ddots ...
1
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3answers
331 views

Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$

Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.
2
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1answer
68 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...